Def. Singular point (of an
analytic function). A point at
which an analytic function f(z) is not
analytic, i.e. at which f '(z) fails to exist, is
called a singular point or singularity of
the function.

There are different types of singular points:

Isolated and non-isolated
singular points. A singular point z0
is called an isolated singular point of an
analytic function f(z) if there exists a
deleted ε-spherical neighborhood of z0 that
contains no singularity. If no such
neighborhood can be found, z0 is called a
non-isolated singular point. Thus an
isolated singular point is a singular point
that stands completely by itself, embedded
in regular points. See Fig. 1a where z1, z2 and z3 are isolated singular points. Most singular
points are isolated singular points. A non-isolated singular point is a singular point such that
every deleted ε-spherical neighborhood of it contains singular points. See Fig. 1b where z0 is the
limit point of a set of singular points. Isolated singular points include poles, removable
singularities, essential singularities and branch points.

Types of isolated singular points

1. Pole. An isolated singular point z0 such that f(z) can be represented by an expression that is
of the form

where n is a positive integer, f(z) is analytic at z0, and f(z0) ≠ 0. The integer n is called the order
of the pole. If n = 1, z0 is called a simple pole.

Example. The function

has a pole of order 3 at z = 2 and simple poles at z = -3 and z = 2.

Shown in Fig. 2 is a modulus
surface of the function f(z) = 1/(z-a) defined on a region R. One sees
the “pole” arising above point a in
the complex plane. Thus the
reason for the term “pole”. A
modulus surface is obtained by
affixing a Z axis to the z plane and
plotting Z = |f(z)| [i.e. plotting the
modulus of f(z)].

2. Removable singular
point. An isolated singular point
z0 such that f can be defined, or
redefined, at z0 in such a way as to
be analytic at z0. A singular point
z0 is removable if
exists.

Example. The singular point z = 0 is a removable singularity of f(z) = (sin z)/z since

3. Essential singular point. A singular point that is not a pole or removable singularity is
called an essential singular point.

Example. f(z) = e 1/(z-3) has an essential singularity at z = 3.

Singular points at infinity. The type of singularity of f(z) at z = ∞ is the same as that
of f(1/w) at w = 0. Consult the following example.

Example. The function f(z) = z2 has a pole of order 2 at z = ∞, since f(1/w) has a pole of order
2 at w = 0.

Using the transformation w = 1/z the point z = 0 (i.e. the origin) is mapped into w = ∞, called the
point at infinity in the w plane. Similarly, we call z = ∞ the point at infinity in the z plane.
To consider the behavior of f(z) at z = ∞, we let z = 1/w and examine the behavior of f(1/w) at w
= 0.